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Physical Review A

Finite-temperature T matrix in a real-time formalism

V. Canivell, L. Garrido, Maxi San Miguel, Pere Seglar

Published Jan 1, 1978• 5 citations
Publisher

Abstract

A generalized off-shell unitarity relation for the two-body scattering $T$ matrix in a many-body medium at finite temperature is derived, through a consistent real-time perturbation expansion by means of Feynman diagrams. We comment on perturbation schemes at finite temperature in connection with an erroneous formulation of the Dyson equation in a paper recently published.

DOI
10.1103/physreva.17.480

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A generalized off-shell unitarity relation for the two-body scattering TTT matrix in a many-body medium at finite temperature is derived, through a consistent real-time perturbation expansion by means of Feynman diagrams. We comment on perturbation schemes at finite temperature in connection with an erroneous formulation of the Dyson equation in a paper recently published.
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
We present a Green's-function formalism generalized to treat arbitrary time-dependent systems initially prepared with an arbitrary density matrix. On the basis of a simultaneous expansion of the usual real-time development operator and the density matrix, we define a Green's function ordered along an extended contour. In this way we are able to treat real-time many-particle Green's functions and initial correlations on the same footing. Both can be decomposed by Wick's theorem. Thus we may utilize familiar diagrammatic analysis for evaluating the Green's function, and finally arrive at a generalized Dyson's equation. The Green's function can be represented by a (3\ifmmode\times\else\texttimes\fi{}3) matrix, which includes the Green's functions used in the Feynman, Matsubara, and Keldysh theories, if the corresponding statistical average is taken. From a matrix representation of Dyson's equation, a basic set of five coupled equations is derived, which explicitly shows the corrections to the Keldysh theory due to an arbitrary initial many-particle density matrix.
The first correction in the S-matrix expansion possesses quite an obvious physical interpretation, because it directly involves the asymptotic conditions in the scattering experiment. The physical meaning of the appearance of the times t 1 and t 2 consists of successive inclusion of the interactions. A physical meaning of such a development is customarily established by calculating the concrete matrix elements based upon equation. As Feynman pointed out, the graphical presentations are not only convenient symbols but also could be considered as good candidates for an adequate description of realistic physical observables in scattering events. Thus, for example, in field theory, the places where the partial segments of the zig-zag lines stop, in fact, become the media of the physical events, i.e. the energy and impulse are conserved at the vertices.
A formulation of a self-consistent perturbation method is presented, which is convenient to deal with the strongly correlated p-d mixing model. A real-time formulation of thermofield dynamics is used. The method allows one to develop a finite-temperature perturbation theory for operators that do not allow the use of Wick's theorem. By introducing local decomposition of the operators, a diagram method that does not rely on Wick's theorem is constructed. For the p-d mixing model, self-consistent equations for the electron Green's functions are derived.
The paper addresses the many-body quasiparticle dynamics of the Anderson impurity model at finite temperatures in the framework of the equation-of-motion method. We find a new exact identity relating the one-particle and many-particle Green’s functions. Using this identity we present a consistent and general scheme for a construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of Dyson equation. A new approach for the complex expansion for the single-particle propagator in terms of Coulomb repulsion U and hybridization V is proposed. Using the exact identity, the essentially new many-body dynamical solution of SIAM has been derived. This approach offers a new way for the systematic construction of the approximative interpolating dynamical solutions of the strongly correlated electron systems.